#403543

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,15,19)(11,20,18,14)(12,21,26,32)(13,27,30,22)(16,24,33,25)(17,29,34,28)(23,36,39,31)(35,37,40,38);(1,2,4,6)(3,7,8,5)(9,11,14,10,15,18,20,19)(12,21,24,16,26,32,25,33)(13,22,28,17,30,27,29,34)(23,35,37,31,39,40,38,36);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,21)(13,23)(14,24)(15,26)(17,31)(18,32)(19,33)(20,25)(22,35)(27,40)(28,37)(29,38)(30,39)(34,36);(9,13,14,28,15,30,20,29)(10,17,18,27,19,34,11,22)(12,23,24,37,26,39,25,38)(16,31,32,40,33,36,21,35)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310739);